A high order purely frequency-based harmonic balance formulation for continuation of periodic solutions

Combinig the harmonic balance method (HBM) and a continuation method is a well-known technique to follow the periodic solutions of dynamical systems when a control parameter is varied. However, since deriving the algebraic system containing the Fourier coefficients can be a highly cumbersome procedure, the classical HBM is often limited to polynomial (quadratic and cubic) nonlinearities and/or a few harmonics. Several variations on the classical HBM, such as the incremental HBM or the alternating frequency/time domain HBM, have been presented in the literature to overcome this shortcoming. Here, we present an alternative approach that can be applied to a very large class of dynamical systems (autonomous or forced) with smooth equations. The main idea is to systematically recast the dynamical system in quadratic polynomial form before applying the HBM. Once the equations have been rendered quadratic, it becomes obvious to derive the algebraic system and solve it by the so-called ANM continuation technique. Several classical examples are presented to illustrate the use of this numerical approach.Comment: PACS numbers: 02.30.Mv, 02.30.Nw, 02.30.Px, 02.60.-x, 02.70.-

7th International Conference on Nonlinear Vibrations, Localization and Energy Transfer: Extended Abstracts

International audienceThe purpose of our conference is more than ever to promote exchange and discussions between scientists from all around the world about the latest research developments in the area of nonlinear vibrations, with a particular emphasis on the concept of nonlinear normal modes and targeted energytransfer

Nonlinear normal modes of a two degree of freedom oscillator with a bilateral elastic stop

A study of the non linear modes of a two degree of freedom mechanical system with bilateral elastic stop is considered. The issue related to the non-smoothness of the impact force is handled through a regularization technique. In order to obtain the Nonlinear Normal Mode (NNM), the harmonic balance method with a large number of harmonics, combined with the asymptotic numerical method, is used to solve the regularized problem. These methods are present in the software "package" MANLAB. The results are validated from periodic orbits obtained analytically in the time domain by direct integration of the non regular problem. The two NNMs starting respectively from the two linear normal modes of the associated underlying linear system are discussed. The energy-frequency plot is used to present a global vision of the behavior of the modes. The dynamics of the modes are also analyzed comparing each periodic orbits and modal lines. The first NNM shows an elaborate dynamics with the occurrence of multiple impacts per period. On the other hand, the second NNM presents a more simple dynamics with a localization of the displacement on the first mass

Towards a full higher order AD-based continuation and bifurcation framework

International audienceSome of the theoretical aspects of continuation and bifurcation methods devoted to the solution for nonlinear parametric systems are presented in a higher-order automatic differentiation (HOAD) framework. Besides benefits in terms of generality and ease of use, HOAD is used to assess fold and simple bifurcations points. In particular, the formation of a geometric series in successive Taylor coefficients allows for the implementation of an efficient detection and branch switching method at simple bifurcation points. Some comparisons with the Auto and MatCont continuation software are proposed. Strengths are then exemplified on a classical case study in structural mechanics

Méthode Asymptotique Numérique adaptative pour la dynamique transitoire non-linéaire

National audienceCet article décrit le couplage de la Méthode Asymptotique Numérique (MAN) avec un schéma d’intégration temporelle permettant la dissipation numérique des hautes fréquences tout en conservant les moments linéaires et angulaires. Différentes stratégies sont présentées afin d’optimiser l’efficacité de la méthode proposée, dont l’adaptation de l’ordre des séries de la MAN à chaque pas de temps. L’algorithme obtenu est appliqué à des structures minces fortement non linéaires discrétisées par éléments finis

Direct computation of paths of limit points using the Asymptotic Numerical Method

International audienceThis paper is concerned with parameter dependent problems for structural instability. The aim is the direct determination of the so called fold curve connecting the limit points of the equilibrium path for a structure subjected to a variable imperfection. This is traditionally achieved by adding a well-chosen constraint equation requiring the criticality of the equilibrium. The crucial feature of the paper lies in the numerical resolution of the obtained augmented system. Indeed, it is solved using the Asymptotic Numerical Method (A.N.M.) which is well-known for its robustness. The theoretical framework upon which the A.N.M. and the extended system are based are presented. From a numerical point of view, it leads to an efficient treatment which takes the singularity of the tangent stiffness matrix into account. Emphasis is given on two specific types of geometrical imperfections. Eventually, the numerical isolation of an initial starting limit point is discussed

Buckling of Imperfect Elastic Shells using the Asymptotic Numerical Method

International audienceThis paper is concerned with stability behaviour and imperfection sensitivity of elastic shells. The aim is to determine the reduction of the critical buckling load as a function of the imperfection amplitude. For this purpose, the direct calculation of the so-called fold line connecting all the limit points of the equilibrium branches of the imperfect structures is performed. An augmented system demanding the criticality of the equilibrium is used. In order to solve the augmented system, the Asymptotic Numerical Method is used as an alternative to Newton-like incremental-iterative procedures. It results in a very robust and efficient path-following algorithm that takes the singularity of the tangent stiffness matrix into account. Two specific types of imperfections are detailed and several numerical examples are discussed

Regards monastiques sur la double clôture de Cluny (xie siècle)

Ma contribution d’historienne à ce volume, rassemblant avant tout des archéologues, est d’analyser un groupe de textes pour essayer de comprendre, à partir de cet échantillon, comment, sur une période et en un lieu donné, le monde monastique médiéval concevait les barrières physiques qui encerclaient le monastère et si cette conception a changé au fil du temps. Cet article explore ce que les moines disaient de la clôture à Cluny au xie siècle dans les coutumiers décrivant la vie de l’abbaye. ..

Low-dimensional Nonlinear Modes computed with PGD/HBM and Reduced Nonlinear Modal Synthesis for Forced Responses

International audienceThis work proposes an algorithm allowing to perform a fast and light computation of branches of damped Nonlinear Normal Modes (dNNMs). Based on a previous work about undamped NNMs (uNNMs), it couples Proper Generalized Decomposition (PGD) features, harmonic balance and prediction-correction continuation schemes. After recalling the main contributions of the method applied on an example with cubic nonlinearities, the issue of a reduced nonlinear modal synthesis is briefly addressed